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  4. Particles of masses m, 2m, 3m, ... , nm grams are placed on the same line at distances l, 2l, 3l, ..., nl cm from a fixed point. The distance of centre of mass of the particles from the fixed point in centimetres is

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Q. Particles of masses $m, 2m, 3m, ... , nm$ grams are placed on the same line at distances $l, 2l, 3l, ..., nl$ cm from a fixed point. The distance of centre of mass of the particles from the fixed point in centimetres is

JIPMERJIPMER 2015System of Particles and Rotational Motion

Solution:

$X_{CM} =\frac{m_{1}x_{1} +m_{2}x_{2}+...}{m_{1}+m_{2}+...}$
$= \frac{ml+2m.2l+3m.3l+...}{m+2m+3m+....}$
$ =\frac{ml\left(1+4+9+...\right)}{m\left(1+2+3+....\right)}$
$=\frac{l \frac{n\left(n+1\right)\left(2n+1\right)}{6}}{\frac{n\left(n+1\right)}{2}}=\frac{l\left(2n+1\right)}{3}$